
where \({\bf v}^+ = \max ({\bf v}, 0)\) , \({\bf v}^- = \min ({\bf v}, 0)\)  and the inner product \(\langle \cdot , \cdot \rangle \)  is in the sense of (REF ).
Let \(F^{n, \pm }_j ={\bf F}^{n, \pm }(j)\) , the upwind fluxes \(\hat{F}^{n, \pm }_{j+\frac{1}{2}}\)  are reconstructed from \({\bf F}^{n, \pm }(:)\)  in the following way using the corresponding high order upwind stencils [1]},
\(\hat{F}^{n, -}_{j+\frac{1}{2}} &= -\frac{1}{20}F_{j-1}^{n, -} + \frac{9}{20}F_{j}^{n, -} + \frac{47}{60}F_{j+1}^{n, -} - \frac{13}{60}F_{j+2}^{n, -}+ \frac{1}{30}F_{j+3}^{n, -},\\\hat{F}^{n, +}_{j+\frac{1}{2}} &= \frac{1}{30}F_{j-2}^{n, +} -\frac{13}{60}F_{j-1}^{n, +} + \frac{47}{60}F_{j}^{n, +} + \frac{9}{20}F_{j+1}^{n, +} -\frac{1}{20}F_{j+2}^{n, +}.\) 
